Sustainability and Stability - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 809

and x¼x 0 satisfies

0 ¼ AxþB (16.26)

ie, x 0 is the steady-state solution of (16.25) . The initial condition is given as
t ¼ 0; x ¼ x I

(16.27)

which defines the state the system is at when time t is zero.
The solution to the set of differential equations is stable if for any give finitely small
numbers and d , with the norm of the difference between the initial conditions and the

steady state solutions to be less than d or kx I e x 0 k < d ,
kxx 0 k < ε

ε

(16.28)

holds for any t. That is, if the solution is bounded to be of finite values, the system is stable.
Furthermore, we say the (steady state) solution x 0 is asymptotically stable if for a given value
of d 0 >

0,

kx I x 0 k < d 0 (16.29)

the following condition holds

lim
t

ðxx 0 Þ¼0 (16.30)

/N

Whether the solution to the set of differential equations is stable or not, the coefficient
matrix A plays a key role. In particular, the real part of all the Eigen values of A must
be < 0, i.e. Re( l ) < 0. The Eigen values ( l 's) can be determined by setting

detðAlEÞ¼0 (16.31)
For example, for a set of two differential equations, there are two Eigen values. The
characteristics of the Eigen values determine how the solution approaches the steady-state
solution. Fig. 16.11 shows the trajectories to the steady-state solution for different combi-
nations of Eigen values. On the (x 1 , x 2 ) plane, we have plotted the lines for steady-state
behaviors ( f 1 ¼ 0 and f 2 ¼ 0). One can observe that if the two Eigen values are real, the
solution approaches the steady-state solution (x 10 , x 20 ) only when both Eigen values are
less than zero. Even then, the values of x 1 and x 2 can oscillate around the steady-state
values (causing overshoot). If the two Eigen values are of conjugate imaginary values,
the solution only approaches the steady-state solution when the real part of the Eigen
values is less than zero. The oscillation is more pronounced when the Eigen values are
not of real values.

Example 16-4 A cell culture grows on a glucose media is inhibited by the substrate under
isothermal conditions and the specific growth rate is governed by

m max S
K S þSþS 2

m G ¼ (E16-4.1)

= K I

m max ¼ 0.1 h 1 , K S ¼ 2 g/L, K I ¼ 25 g/L. The cells has a specific death rate of k d ¼ 0.01/h
and a yield factor of YF X/S ¼ 0.8. The culture is grown in a chemostat with a dilution rate of
D ¼ 0.04 h 1 . Initially, the feed was sterile containing S 0I ¼ 100 g/L substrate (Example 16-2).

with